These results reflect my experience with this sort of problem.
A couple of comments:
1) optimx package has a multistart wrapper. I probably should have written one
for
nlsr. Maybe Bernard and I should work on that. The issues are largely to make
things
resistant to silly inputs, which even the smart users (you know, the ones
looking
back from the mirror) introduce.
2) Sometimes using the bounds constraint capability in nlsr can be helpful,
e.g.,
to ensure the exponent parameters are kept apart, can be useful.
3) Combining with Duncan's suggestion of solving for the linear parameters also
helps.
All of the above can be sensitive to particular data.
Best, JN
On 2020-05-07 5:41 p.m., Bernard McGarvey wrote:
> John/Petr, I think there is an issue between a global optimum and local
> optima. I added a multistart loop around the code to see if I could find
> different solutions. Here is the code I added (I am not a great coder so
> please excuse any inefficiencies in this code segment):
>
> # Multistart approach
> NT <- 100
> Results <- matrix(data=NA, nrow = NT, ncol=5, dimnames=list(NULL,c("SS", "A",
> "B", "a", "b")))
> A1 <- runif(NT,0,100)
> B1 <- runif(NT,0,100)
> a1 <- runif(NT,0.0,0.1)
> b1 <- runif(NT,0.0,0.1)
> for (I in 1:NT) {
> if (A1[I] > B1[I]) { # Ensure that the A'a are always the lower so that
> nlxb() always converge to the same values
> A0 <- A1[I]
> a0 <- a1[I]
> A1[I] <- B1[I]
> a1[I] <- b1[I]
> B1[I] <- A0
> b1[I] <- a0
> }
> fit <- nlxb(tsmes ~ A*exp(a*plast) + B* exp(b*plast), data=temp,
> start=list(A=A1[I], B=B1[I], a=a1[I], b=b1[I]))
> ccc <- coef(fit)
> Results[I,1] <- fit$ssquares
> Results[I,2] <- ccc[1]
> Results[I,3] <- ccc[2]
> Results[I,4] <- ccc[3]
> Results[I,5] <- ccc[4]
> }
> Results
>
> What I found is that the minimum SS generated at each trial had two distinct
> values, 417.8 and 3359.2. The A,B,a, and b values when the SS was 417.8 were
> all the same but I got different values for the case where the minimal SS was
> 3359.2. This indicates that the SS=417.8 may be the global minimum solution
> whereas the others are local optima. Here are the iteration results for a 100
> trial multistart:
>
> Results
> SS A B a b
> [1,] 3359.2 8.3546e+03 6.8321e+00 -1.988226 2.6139e-02
> [2,] 3359.2 8.2865e+03 6.8321e+00 -5.201735 2.6139e-02
> [3,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [4,] 3359.2 6.8321e+00 7.7888e+02 0.026139 -7.2812e-01
> [5,] 3359.2 -3.9020e+01 4.5852e+01 0.026139 2.6139e-02
> [6,] 3359.2 6.8321e+00 2.6310e+02 0.026139 -1.8116e+00
> [7,] 3359.2 -2.1509e+01 2.8341e+01 0.026139 2.6139e-02
> [8,] 3359.2 -3.8075e+01 4.4908e+01 0.026139 2.6139e-02
> [9,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [10,] 3359.2 1.2466e+04 6.8321e+00 -4.196000 2.6139e-02
> [11,] 417.8 9.7727e+00 3.9452e-13 0.021798 2.8023e-01
> [12,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [13,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [14,] 3359.2 3.8018e+02 6.8321e+00 -0.806414 2.6139e-02
> [15,] 3359.2 -3.1921e+00 1.0024e+01 0.026139 2.6139e-02
> [16,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [17,] 3359.2 -1.5938e+01 2.2770e+01 0.026139 2.6139e-02
> [18,] 3359.2 -3.1205e+01 3.8037e+01 0.026139 2.6139e-02
> [19,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [20,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [21,] 3359.2 8.6627e+03 6.8321e+00 -3.319778 2.6139e-02
> [22,] 3359.2 6.8321e+00 1.9318e+01 0.026139 -6.5773e-01
> [23,] 3359.2 6.2991e+01 -5.6159e+01 0.026139 2.6139e-02
> [24,] 3359.2 2.8865e-03 6.8321e+00 -1.576307 2.6139e-02
> [25,] 3359.2 -1.2496e+01 1.9328e+01 0.026139 2.6139e-02
> [26,] 3359.2 -5.9432e+00 1.2775e+01 0.026139 2.6139e-02
> [27,] 3359.2 1.6884e+02 6.8321e+00 -211.866423 2.6139e-02
> [28,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [29,] 3359.2 5.4972e+03 6.8321e+00 -3.432094 2.6139e-02
> [30,] 3359.2 6.8321e+00 1.4427e+03 0.026139 -4.2771e+02
> [31,] 417.8 9.7727e+00 3.9452e-13 0.021798 2.8023e-01
> [32,] 3359.2 3.5760e+01 -2.8928e+01 0.026139 2.6139e-02
> [33,] 3359.2 6.8321e+00 -4.0737e+02 0.026139 -6.7152e-01
> [34,] 3359.2 6.8321e+00 1.2638e+04 0.026139 -2.8070e+00
> [35,] 3359.2 1.1813e+01 -4.9807e+00 0.026139 2.6139e-02
> [36,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [37,] 3359.2 6.8321e+00 1.2281e+03 0.026139 -3.0702e+02
> [38,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [39,] 3359.2 -2.6850e+01 3.3682e+01 0.026139 2.6139e-02
> [40,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [41,] 417.8 9.7727e+00 3.9452e-13 0.021798 2.8023e-01
> [42,] 3359.2 -2.3279e+01 3.0111e+01 0.026139 2.6139e-02
> [43,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [44,] 3359.2 6.8321e+00 1.4550e+03 0.026139 -4.0303e+00
> [45,] 3359.2 -1.1386e+01 1.8218e+01 0.026139 2.6139e-02
> [46,] 3359.2 8.8026e+02 6.8321e+00 -65.430608 2.6139e-02
> [47,] 3359.2 -8.1985e+00 1.5031e+01 0.026139 2.6139e-02
> [48,] 3359.2 -6.7767e+00 1.3609e+01 0.026139 2.6139e-02
> [49,] 3359.2 -1.1436e+01 1.8268e+01 0.026139 2.6139e-02
> [50,] 3359.2 1.0710e+04 6.8321e+00 -2.349659 2.6139e-02
> [51,] 417.8 9.7727e+00 3.9452e-13 0.021798 2.8023e-01
> [52,] 3359.2 6.8321e+00 7.1837e+02 0.026139 -7.4681e-01
> [53,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [54,] 417.8 9.7727e+00 3.9452e-13 0.021798 2.8023e-01
> [55,] 3359.2 -4.8774e+00 6.8321e+00 -16.405584 2.6139e-02
> [56,] 3359.2 1.2687e+03 6.8321e+00 -3.775998 2.6139e-02
> [57,] 3359.2 1.5529e+01 -8.6967e+00 0.026139 2.6139e-02
> [58,] 3359.2 -1.0003e+01 1.6835e+01 0.026139 2.6139e-02
> [59,] 3359.2 6.8321e+00 3.9291e+02 0.026139 -4.1974e+02
> [60,] 3359.2 -2.1880e+01 2.8712e+01 0.026139 2.6139e-02
> [61,] 3359.2 4.1736e+03 6.8321e+00 -10.711457 2.6139e-02
> [62,] 3359.2 -3.3185e+01 4.0017e+01 0.026139 2.6139e-02
> [63,] 3359.2 7.6732e+02 6.8321e+00 -0.723977 2.6139e-02
> [64,] 3359.2 1.5334e+04 6.8321e+00 -52.573620 2.6139e-02
> [65,] 3359.2 -2.9556e+01 3.6388e+01 0.026139 2.6139e-02
> [66,] 3359.2 -1.0447e+00 7.8767e+00 0.026139 2.6139e-02
> [67,] 3359.2 6.8321e+00 2.1471e+02 0.026139 -7.0582e+01
> [68,] 417.8 9.7727e+00 3.9452e-13 0.021798 2.8023e-01
> [69,] 3359.2 -2.2293e+01 2.9126e+01 0.026139 2.6139e-02
> [70,] 3359.2 6.2259e+02 6.8321e+00 -2.782527 2.6139e-02
> [71,] 3359.2 -1.4639e+01 2.1471e+01 0.026139 2.6139e-02
> [72,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [73,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [74,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [75,] 3359.2 -2.3449e+01 3.0281e+01 0.026139 2.6139e-02
> [76,] 3359.2 -2.5926e+01 6.8321e+00 -0.663656 2.6139e-02
> [77,] 417.8 9.7727e+00 3.9452e-13 0.021798 2.8023e-01
> [78,] 3359.2 6.8321e+00 6.9426e+02 0.026139 -1.9442e+00
> [79,] 3359.2 2.8684e+02 6.8321e+00 -0.854394 2.6139e-02
> [80,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [81,] 3359.2 -4.5066e+01 5.1899e+01 0.026139 2.6139e-02
> [82,] 3359.2 4.4678e+03 6.8321e+00 -2.109446 2.6139e-02
> [83,] 3359.2 3.1376e+03 6.8321e+00 -1.104803 2.6139e-02
> [84,] 3359.2 6.8321e+00 1.1167e+02 0.026139 -1.0280e+00
> [85,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [86,] 3359.2 5.3864e+02 6.8321e+00 -0.657971 2.6139e-02
> [87,] 3359.2 4.8227e+01 6.8321e+00 -2.304024 2.6139e-02
> [88,] 3359.2 -2.2048e+01 2.8880e+01 0.026139 2.6139e-02
> [89,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
> [90,] 3359.2 6.8321e+00 -4.1689e+01 0.026139 -3.6049e+00
> [91,] 417.8 9.7727e+00 3.9452e-13 0.021798 2.8023e-01
> [92,] 3359.2 -4.1265e+01 4.8097e+01 0.026139 2.6139e-02
> [93,] 3359.2 -1.1565e+01 1.8397e+01 0.026139 2.6139e-02
> [94,] 3359.2 2.3698e+01 -1.6866e+01 0.026139 2.6139e-02
> [95,] 3359.2 4.4700e+03 6.8321e+00 -12.836180 2.6139e-02
> [96,] 3359.2 4.6052e+04 6.8321e+00 -7.158584 2.6139e-02
> [97,] 3359.2 2.5464e+03 6.8321e+00 -1.811626 2.6139e-02
> [98,] 3359.2 6.8321e+00 1.0338e+03 0.026139 -1.5365e+01
> [99,] 3359.2 1.3783e+01 -6.9507e+00 0.026139 2.6139e-02
> [100,] 3359.2 6.8321e+00 6.7153e+02 0.026139 -1.5975e+03
>
>
> Hope this helps,
>
> Bernard McGarvey
>
>
> Director, Fort Myers Beach Lions Foundation, Inc.
>
>
> Retired (Lilly Engineering Fellow).
>
>> On May 7, 2020 at 9:33 AM J C Nash <profjcn...@gmail.com> wrote:
>>
>>
>> The double exponential is well-known as a disaster to fit. Lanczos in his
>> 1956 book Applied Analysis, p. 276 gives a good example which is worked
>> through.
>> I've included it with scripts using nlxb in my 2014 book on Nonlinear
>> Parameter
>> Optimization Using R Tools (Wiley). The scripts were on Wiley's site for the
>> book,
>> but I've had difficulty getting Wiley to fix things and not checked lately
>> if it
>> is still accessible. Ask off-list if you want the script and I'll dig into my
>> archives.
>>
>> nlxb (preferably from nlsr which you used rather than nlmrt which is now not
>> maintained), will likely do as well as any general purpose code. There may be
>> special approaches that do a bit better, but I suspect the reality is that
>> the underlying problem is such that there are many sets of parameters with
>> widely different values that will get quite similar sums of squares.
>>
>> Best, JN
>>
>>
>> On 2020-05-07 9:12 a.m., PIKAL Petr wrote:
>>> Dear all
>>>
>>> I started to use nlxb instead of nls to get rid of singular gradient error.
>>> I try to fit double exponential function to my data, but results I obtain
>>> are strongly dependent on starting values.
>>>
>>> tsmes ~ A*exp(a*plast) + B* exp(b*plast)
>>>
>>> Changing b from 0.1 to 0.01 gives me completely different results. I usually
>>> check result by a plot but could the result be inspected if it achieved good
>>> result without plotting?
>>>
>>> Or is there any way how to perform such task?
>>>
>>> Cheers
>>> Petr
>>>
>>> Below is working example.
>>>
>>>> dput(temp)
>>> temp <- structure(list(tsmes = c(31, 32, 32, 32, 32, 32, 32, 32, 33,
>>> 34, 35, 35, 36, 36, 36, 37, 38, 39, 40, 40, 40, 40, 40, 41, 43,
>>> 44, 44, 44, 46, 47, 47, 47, 47, 48, 49, 51, 51, 51, 52, 53, 54,
>>> 54, 55, 57, 57, 57, 59, 59, 60, 62, 63, 64, 65, 66, 66, 67, 67,
>>> 68, 70, 72, 74, 76, 78, 81, 84, 85, 86, 88, 90, 91, 92, 94, 96,
>>> 97, 99, 100, 102, 104, 106, 109, 112, 115, 119, 123, 127, 133,
>>> 141, 153, 163, 171), plast = c(50, 51, 52, 52, 53, 53, 53, 54,
>>> 55, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 64, 64, 65, 65, 66,
>>> 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 75, 76, 76, 77, 77, 78,
>>> 78, 79, 80, 81, 82, 83, 84, 85, 85, 86, 86, 87, 88, 88, 89, 90,
>>> 91, 91, 93, 93, 94, 95, 96, 96, 97, 98, 98, 99, 100, 100, 101,
>>> 102, 103, 103, 104, 105, 106, 107, 107, 108, 109, 110, 111, 112,
>>> 112, 113, 113, 114, 115, 116)), row.names = 2411:2500, class = "data.frame")
>>>
>>> library(nlsr)
>>>
>>> fit <- nlxb(tsmes ~ A*exp(a*plast) + B* exp(b*plast), data=temp,
>>> start=list(A=1, B=15, a=0.025, b=0.01))
>>> coef(fit)
>>> A B a b
>>> 3.945167e-13 9.772749e+00 2.802274e-01 2.179781e-02
>>>
>>> plot(temp$plast, temp$tsmes, ylim=c(0,200))
>>> lines(temp$plast, predict(fit, newdata=temp), col="pink", lwd=3)
>>> ccc <- coef(fit)
>>> lines(0:120,ccc[1]*exp(ccc[3]*(0:120)))
>>> lines(0:120,ccc[2]*exp(ccc[4]*(0:120)), lty=3, lwd=2)
>>>
>>> # wrong fit with slightly different b
>>> fit <- nlxb(tsmes ~ A*exp(a*plast) + B* exp(b*plast), data=temp,
>>> start=list(A=1, B=15, a=0.025, b=0.1))
>>> coef(fit)
>>> A B a b
>>> 2911.6448377 6.8320597 -49.1373979 0.0261391
>>> lines(temp$plast, predict(fit, newdata=temp), col="red", lwd=3)
>>> ccc <- coef(fit)
>>> lines(0:120,ccc[1]*exp(ccc[3]*(0:120)), col="red")
>>> lines(0:120,ccc[2]*exp(ccc[4]*(0:120)), lty=3, lwd=2, col="red")
>>>
>>>
>>> ______________________________________________
>>> R-help@r-project.org mailing list -- To UNSUBSCRIBE and more, see
>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
>>> and provide commented, minimal, self-contained, reproducible code.
>>>
>>
>> ______________________________________________
>> R-help@r-project.org mailing list -- To UNSUBSCRIBE and more, see
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.
______________________________________________
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